A.A. 2018/2019

Insegnamenti attivati

  • Analisi complessa e funzionale - Functional Analysis and Complex Analysis
    Nicola Arcozzi con Pavel Mozolyako
    corso concluso
  • Integrazione gaussiana e meccanica statistica
    Pierluigi Contucci
    15 h
  • Introduzione alla geometria di Cartan e sue applicazioni
    Emanuele Latini – Andrea Santi
    corso concluso
  • Ottimizzazione convessa - Convex Optimization for Imaging
    Alessandro Lanza
    orario:
    10 Aprile (mercoledì) 14-16

    11 Aprile (giovedì) 14-16
    15 Aprile (lunedì) 14-16
    17 Aprile (mercoledì) 14-16
    29 Aprile (lunedì) 14-16
    6  Maggio (lunedì) 14-16
    7  Maggio (martedì) 9-12
    programma:
    In this course, we give an overview of convex optimization methods for signal/image processing applications.
    Topics which we would like to cover:
    - Variational methods for Imaging
    -     Review on standard tools in convex optimization such as, e.g., (strong) convexity, sub-differentials, gradient descent.
    - Proximal methods for non-smooth optimization (acceleration à la Nesterov)
    -     Composite optimization: forward-backward splitting, accelerated forward-backward, FISTA
    -     Alternating Direction Method of Multipliers (ADMM) and Majorize-Minimize approach.
    -     Numerical implementation and simulations in MATLAB for exemplar imaging problems.

Seminari o cicli seminariali

  • Topics in Math
    Organizzatori: Giovanni Cupini e Giovanni Mongardi
    20 h

 

  • Metodi variazionali e PDE per l’elaborazione delle immagini
    Luca Calatroni
    orario:
    dalle 10 alle 12 in Seminario I nei giorni 9, 10, 13, 14, 15, 16 e 17 maggio
    dalle 14 alle 15 in Seminatio I il 15 maggio.
    In this course we will present some classical and recent approaches for some problems in image reconstruction (denoising, deblurring, inpainting, shadow-removal…) formulated in terms of appropriate minimisation problems in infinite-dimensional functional spaces. We will further draw connections between these minimisation problems and parabolic Partial Differential Equations (PDEs) based on non-linear diffusion and possibly combined with transport terms.
    For the practical implementation of the models above, we will review standard finite difference stencils discussing their extensions to anisotropic diffusion and diffusion-transport problems. The course will be complemented by some practical MATLAB classes where simple exemplar problems will be solved by means of some reference iterative algorithms.
    • Classical examples of imaging problems (denoising, deblurring, inpainting, segmentation..). Formulation as ill-posed inverse problems. Variational regularisation methods: regularisation term VS data fitting. Statistical interpretation: MAP estimation (2h)
    • Sobolev spaces, standard methods in calculus of variations: a review. Total variation, the space of functions of bounded variations (2h)
    • Second-order parabolic PDEs for image processing: heat equation, mean-curvature flow. Applications to image processing: linear VS non-linear PDEs. Regularisation of non-smoothness: lagged diffusivity. Anisotropic diffusion and diffusion-transport problems. (4h)
    • Finite differences stencils for PDE-based imaging models. (2h)
    • Numerical implementation and simulations in MATLAB for PDE-based models for image reconstruction (deblurring, inpainting, face fusion). (5h)
  • Introduction to scattering resonances
    Maciej Zworski
    orario:
    3, 4, 6 e 7 giugno dalla 10 alle 13
    Scattering resonances replace bound states/eigenvalues for spectral problems in which escape (scattering) to infinity is possible. These states have rates of oscillation and decay and that information is elegantly encoded in considering the corresponding ``eigenvalues" as poles of the meromorphic continuation of Green functions. The most famous ``pure maths" example is given by zeros of the Riemann zeta function which can be interpreted as resonances for scattering on the modular surface. In ``applied maths" they appear anywhere from gravitational waves to MEMS (Micro-Electro-Mechanical Systems).
    The mini course will provide a gentle introduction in the setting of potential scattering in dimension three. Only basic functional analysis will be a prerequisite.
    1. One dimensional scattering: intuition behind outgoing and incoming waves and the definition of scattering resonances.
    2. Analytic Fredholm theory and, as application, meromorphic continuation of Green's function for potentials scattering in dimension three.
    3. Resonance free regions and expansion of waves in terms of resonances.
    4. Counting resonances: upper bounds and existence (and some open problems). Complex valued potentials with no resonances.
    Section 2 of https://math.berkeley.edu/~zworski/revres.pdf (Bull Math Sci '17) will provide a reference with a more detailed presentation in the forthcoming book http://math.mit.edu/~dyatlov/res/ (AMS '19, to appear).

  • The de Branges theory of Hilbert spaces of entire functions and its applications to spectral theory of differential operators
    Anton Baranov
    info and schedule

 

  • Topics in global analysis
    Gerardo A. Mendoza
    14 h - seconda metà di maggio

Altre attività didattiche e di ricerca

  • Fruizione dei seminari di Dipartimento e in particolare seminari a scelta su tematiche specifiche per ciascun dottorando come: seminari di Algebra e Geometria, Seminario Pini, AM2, Fisica Matematica.
    Sito web